handout 9 2018 - forsiden - universitetet i oslo · 14.10.2018 5 «consider the components of the...
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Particle Physics
Handout 9 : The Weak Interaction and V-A
FYS4555 – UiO – 2018 Based on Prof Mark Thomson’s slides
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Parity«The parity operator performs spatial inversion through the origin:
• To preserve the normalisation of the wave-function
Unitary
• applying twice:
• But since Hermitianwhich implies Parity is an observable quantity. If the interaction Hamiltoniancommutes with , parity is an observable conserved quantity
• If is an eigenfunction of the parity operator with eigenvalue
since Parity has eigenvalues
so
« QED and QCD are invariant under parity« Experimentally observe that Weak Interactions do not conserve parity
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Intrinsic Parities of fundamental particles:
• From the Dirac equation showed (handout 2):Spin ½ particles have opposite parity to spin ½ anti-particles
• Conventional choice: spin ½ particles have
and anti-particles have opposite parity, i.e.
Spin-½ Fermions
Spin-1 Bosons
• From Gauge Field Theory can show that the gauge bosons have
« For Dirac spinors it was shown (handout 2) that the parity operator is:
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Parity Conservation in QED and QCD
e–e–
q q
• The Feynman rules for QED give:
• Which can be expressed in terms of the electron andquark 4-vector currents:
• Consider the QED process e–q ¦ e–q
with and
«Consider what happens to the matrix element under the parity transformations Spinors transform as
s Adjoint spinors transform as
s Hence
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« Consider the components of the four-vector current
• The time-like component remains unchanged and the space-like componentschange sign
since
since
0:
k=1,2,3:
• Similarly « Consequently the four-vector scalar product
QED Matrix Elements are Parity Invariant
Parity Conserved in QED
« The QCD vertex has the same form and, thus,
Parity Conserved in QCD
or
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Parity Violation in b-Decay
Vectorschange sign
Axial-Vectorsunchanged
«1957: C.S.Wu et al. studied beta decay of polarized cobalt-60 nuclei:
«Under the parity transformation:
«Observed electrons emitted preferentially in direction opposite to applied field
If parity were conserved: expect equal rate for producing e– in directions along and opposite to the nuclear spin.
«The parity operator corresponds to a discrete transformation
more e- in c.f.
Note B is anaxial vector
«Conclude parity is violated in WEAK INTERACTIONthat the WEAK interaction vertex is NOT of the form
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Bilinear Covariants«The requirement of Lorentz invariance of the matrix element severely restricts
the form of the interaction vertex. QED and QCD are “VECTOR” interactions:
«This combination transforms as a 4-vector (Handout 2 extra V)« In general, there are only 5 possible combinations of two spinors and the gamma
matrices that form Lorentz invariant currents, called “bilinear covariants”:
« Note that in total the sixteen components correspond to the 16 elements ofa general 4x4 matrix: “decomposition into Lorentz invariant combinations”
« In QED the factor arose from the sum over polarization states of the virtualphoton (2 transverse + 1 longitudinal, 1 scalar) = (2J+1) + 1
s SCALAR s PSEUDOSCALAR s VECTORs AXIAL VECTORs TENSOR
Type Form Components “Boson Spin”1 1 446
0 0 112
« Associate SCALAR and PSEUDOSCALAR interactions with the exchange of a SPIN-0 boson, etc. – no spin degrees of freedom
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V-A Structure of the Weak Interaction«The most general form for the interaction between a fermion and a boson is a
linear combination of bilinear covariants« For an interaction corresponding to the exchange of a spin-1 particle the most
general form is a linear combination of VECTOR and AXIAL-VECTOR«The form for WEAK interaction is determined from experiment to be
VECTOR – AXIAL-VECTOR (V – A)
e– ne
V – A« Can this account for parity violation?« First consider parity transformation of a pure AXIAL-VECTOR current
with
or
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• The space-like components remain unchanged and the time-like componentschange sign (the opposite to the parity properties of a vector-current)
• Now consider the matrix elements
• For the combination of a two axial-vector currents
• Consequently parity is conserved for both a pure vector and pure axial-vector interactions
• However the combination of a vector current and an axial vector current
changes sign under parity – can give parity violation !
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« Now consider a general linear combination of VECTOR and AXIAL-VECTOR
(note this is relevant for the Z-boson vertex)
• Consider the parity transformation of this scalar product
• If either gA or gV is zero, Parity is conserved, i.e. parity conserved in a
pure VECTOR or pure AXIAL-VECTOR interaction
• Relative strength of parity violating part
Maximal Parity Violation for V-A (or V+A)
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Chiral Structure of QED (Reminder)« Recall (Handout 4) introduced CHIRAL projections operators
project out chiral right- and left- handed states« In the ultra-relativistic limit, chiral states correspond to helicity states« Any spinor can be expressed as:
• The QED vertex in terms of chiral states:
conserves chirality, e.g.
« In the ultra-relativistic limit only two helicity combinations are non-zero
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Helicity Structure of the WEAK Interactione– ne«The charged current (W±) weak vertex is:
«Since projects out left-handed chiral particle states:
Only the left-handed chiral components of particle spinors and right-handed chiral components of anti-particle spinors participate in charged current weak interactions
«At very high energy , the left-handed chiral components are helicity eigenstates :
LEFT-HANDED PARTICLES Helicity = -1
RIGHT-HANDED ANTI-PARTICLES Helicity = +1
«Writing and from discussion of QED, gives
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In the ultra-relativistic limit only left-handed particles and right-handed antiparticles
participate in charged current weak interactions
e+ nee– ne
e–
nee.g. In the relativistic limit, the only possible electron – neutrino interactions are:
RH anti-particle LH particle RH particle LH anti-particle
« The helicity dependence of the weak interaction parity violation e.g.
Valid weak interaction Does not occur
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Helicity in Pion Decay«The decays of charged pions provide a good demonstration of the role of
helicity in the weak interaction
EXPERIMENTALLY:
• Might expect the decay to electrons to dominate – due to increased phasespace…. The opposite happens, the electron decay is helicity suppressed
«Consider decay in pion rest frame. • Pion is spin zero: so the spins of the n and µ are opposite• Weak interaction only couples to RH chiral anti-particle states. Since
neutrinos are (almost) massless, must be in RH Helicity state • Therefore, to conserve angular mom. muon is emitted in a RH HELICITY state
• But only left-handed CHIRAL particle states participate in weak interaction
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«The general right-handed helicity solution to the Dirac equation is
with and
• project out the left-handed chiralpart of the wave-function using
giving
• similarly
In the limit this tends to zero
In the limit ,
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RH Helicity LH ChiralRH Chiral
• In the limit , as expected, the RH chiral and helicity states are identical • Although only LH chiral particles participate in the weak interaction
the contribution from RH Helicity states is not necessarily zero !
mn ≈ 0: RH Helicity ≡ RH Chiral mµ ≠ 0: RH Helicity has LH Chiral Component
« Expect matrix element to be proportional to LH chiral component of RH Helicityelectron/muon spinor
from the kinematics of pion decay at rest
« Hence because the electron mass is much smaller than the pion mass the decayis heavily suppressed.
«Hence
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Evidence for V-A
e.g. TWIST expt: 6x109 µ decaysPhys. Rev. Lett. 95 (2005) 101805
(solve problems)«The V-A nature of the charged current weak interaction vertex fits with experiment
EXAMPLE charged pion decay
• Experimentally measure:
V-A or V+A
Scalar or Pseudo-Scalar
• Theoretical predictions (depend on Lorentz Structure of the interaction)
EXAMPLE muon decayMeasure electron energy and angular distributions relative to muon spin direction. Results expressed in termsof general S+P+V+A+T form in “Michel Parameters”
V-A Prediction:
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Weak Charged Current Propagator
W-boson propagator
spin 1 W±
«The charged-current Weak interaction is different from QED and QCD in that it is mediated by massive W-bosons (80.3 GeV)
«This results in a more complicated form for the propagator:• in handout 4 showed that for the exchange of a massive particle:
massless massive
• In addition the sum over W boson polarization states modifies the numerator
W-boson propagator ( )
« However in the limit where is small compared with the interaction takes a simpler form.
• The interaction appears point-like (i.e no q2 dependence)
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Connection to Fermi Theory
«In 1934, before the discovery of parity violation, Fermi proposed, in analogy
with QED, that the invariant matrix element for b-decay was of the form:
«After the discovery of parity violation in 1957 this was modified to
(the factor of √2 was included so the numerical value of GF did not need to be changed)
«Compare to the prediction for W-boson exchange
which for becomes:
• Note the absence of a propagator : i.e. this represents an interaction at a point
where
Still usually use to express strength
of weak interaction as it is the quantity
that is precisely determined in muon decay
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Strength of Weak Interaction
« Strength of weak interaction most precisely measured in muon decay
• Here
• To a very good approximation the W-boson
propagator can be written
• In muon decay measure
• Muon decay
« To obtain the intrinsic strength of weak interaction need to know mass of
W-boson: (see handout 14)
The intrinsic strength of the weak interaction is similar to, but greater than,
the EM interaction ! It is the massive W-boson in the propagator which makes
it appear weak. For weak interactions are more likely than EM.
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Summary« Weak interaction is of form Vector – Axial-vector (V-A)
« Consequently only left-handed chiral particle states and right-handedchiral anti-particle states participate in the weak interaction
MAXIMAL PARITY VIOLATION
« At low weak interaction is only weak because of the large W-bosonmass
« Intrinsic strength of weak interaction is similar to that of QED
« Weak interaction also violates Charge Conjugation symmetry
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Solve problems
• 1.1, 11.5, • Calculate the !- decay rate in case of V-A (section 11.6)
and compare ! → #$ /! → &$• 11.6, 11.8, 11.10
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